Instantaneous Acceleration$= \dfrac{dv}{dt} = \dfrac{d^2x}{dt^2}$ … rate of change of velocity with respect to time

Other common phrases

Momentarily at rest means that the object briefly comes to a stop before moving off again.

If the object is momentarily at rest, then the velocity is zero $\dfrac{dx}{dt} = 0$

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Example 1

A particle moves in a straight line such that its position is given by:
… … $x(t) = t^2 - 4t + 3, \qquad t \geqslant 0$

… … where x is measured in metres and t is measured in seconds

… a) .. Find the original position
… b) .. Find the original velocity and describe the initial direction of motion
… c) .. Find the original acceleration
… d) .. Find the time and position when the particle is momentarily at rest
… e) .. Find the position, and velocity after 3 seconds
… f) .. Find the average velocity and average speed over the first three seconds
… g) .. Draw a Position-Time Graph
… h) .. Draw a Velocity-Time Graph

Solution

… a) .. Find the original position

… … … $x(t) = t^2 - 4t + 3$

… … … $x(0) = 3$

… … … initial position at +3m (or 3m right of origin)
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… b) .. Find the original velocity and describe the initial direction of motion

… … … $x(t) = t^2 - 4t + 3$

… … … $v(t) = x'(t) = 2t - 4$

… … … $v(0) = -4$

… … … original velocity is -4m/s (or a speed of 4 m/s to the left)
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… c) .. Find the original acceleration

… … … $v(t) = 2t - 4$

… … … $a(t) = v'(t) = 2$

… … … $a(0) = 2$

… … … acceleration is a constant 2 m/s2 to the right
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… d) .. Find the time and position when the particle is momentarily at rest

… … … $v(t) = 2t - 4$

… … … momentarily at rest when $v(t) = 0$

… … … $2t - 4 = 0$

… … … $t = 2$
.

… … … $x(t) = t^2 - 4t + 3$

… … … $x(2) = -1$

… … … particle is at rest at 2 seconds and is at -1m (1m left of origin)
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… e) .. Find the position, and velocity after 3 seconds

… … … $x(t) = t^2 - 4t + 3$

… … … $x(3) = 0$
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… … … $v(t) = 2t - 4$

… … … $v(3) = 2$

… … … At 3 seconds, particle is at the origin (x = 0) and has velocity 2 m/s to the right
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… f) .. Find the average velocity and average speed over the first three seconds